Analytical approach for the problems of dynamics and stability of a moving web

Abstract

Problems of dynamics and stability of a moving web, modelled as an elastic rod or string, and axially travelling between rollers (supports) at a constant velocity, are studied using analytical approaches. Transverse, longitudinal and torsional vibrations of the moving web are described by a hyperbolic second-order partial differential equation, corresponding to the string and rod models. It is shown that in the framework of a quasi-static eigenvalue analysis, for these models, the critical point cannot be unstable. The critical velocities of one-dimensional webs, and the arising non-trivial solution of free vibrations, are studied analytically. The dynamical analysis is then extended into the case with damping. The critical points of both static and dynamic types are found analytically. It is shown in the paper that if external friction is present, then for mode numbers sufficiently high, dynamic critical points may exist. Graphical examples of eigenvalue spectra are given for both the undamped and damped systems. In the examples, it is seen that external friction leads to stabilization, whereas internal friction in the travelling material will destabilize the system in a dynamic mode at the static critical point. The theory and results summarize and extend theoretical knowledge of the class of models studied, and can be used in various applications of moving materials, such as paper making processes.